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Are polymer chains in blends ”localized” before their phase separation? Variational treatment and prediction
of anomalous scattering results
T. Vilgis, G. Meier
To cite this version:
T. Vilgis, G. Meier. Are polymer chains in blends ”localized” before their phase separation? Varia- tional treatment and prediction of anomalous scattering results. Journal de Physique I, EDP Sciences, 1994, 4 (7), pp.985-990. �10.1051/jp1:1994177�. �jpa-00246967�
Classification Physics Abstracts
05.20 05.70J 61.25H
Short Communication
Are polymer chains in blends "localized" before their phase separation? Variational treatment and prediction of anomalous
scattering results
TA. Vilgis and G. Meier
Max-Planck-Institut für Polymerforschung, Postfach 3148, 55021 Mainz, Germany (Received 15 March 1994, accepted in final form 24 May1994)
Abstract In this communication we suggest, using the droplet picture in phase transitions, that polymer chains are localized before the phase separation in polymer mixtures. The effect is
expected to be significant when the correlation length is of the order of the size of the polymer chain, 1-e- usually for conventional polymers well above the phase separation. Several influences
on experimental results are suggested. The first concerns the dynamic scattering function. This quantity ares net relax to zero, but remains constant for longer times, as already indicated in first experiments. The static structure factor should show a hump in the Kratky plot.
Introduction.
The conformational behaviour of a tagged polymer chain in a binary nlixture of polymers was
studied recently [1]. It was shown that the effective nlonomer-mouomer potential betweeu
nlonomers along the chain is proportionalto the inverse of the collective structure factor and a
prefactor that changes sign before phase separation. It is shown in this paper that this effective poteut1al and its magnitude, given by the correlation leugth, "localizes" individual chains of
the saule species. To treat this problem theoretically we employ the variatioual principle [2],
which was first put forward by Edwards [3] and used very frequently. For short range (excluded volume) potentials the variatioual treatment is known Dot to be reliable [4], whereas for long
range potentials it has beeu round to be useful [Si. As the effective nlonomer potent1al is long ranged (near the phase separation) results fronlthe variational treatment can be expected to be useful.
Recent neutron spin echo expennlents by Ewen et al. [6] on a conlpatible binary polynler blend (d-polyd1nlethylsiloxane, PDMS/polyethylmethylsiloxane, PEMS, with #~,pEMs " 0.49
and a nlean degree of polymerization N
= 330, exhibiting a lower nliscibility gap with a T~ = l10 °C) showed an anonlalous behaviour in the cohereut dyuarnic structure factor S(k,t).
The observed t1nle evolutiou of S(k,t)/S(k,0) did not follow a Rouse like behaviour, as was expected for these polymers since N was below the entanglement nlolecular weight, but re- n1ained nearly constant over a wide range of times [6]. This result showed s1nlilar behaviour to
986 JOURNAL DE PHYSIQUE I N°7
those obtained front a high molecular nlelt of a poly(ethyleue-propylene) alteruatiug copoly-
n1er where the entanglement distance has been observed by neutron spin echo spectroscopy [7]. The intuitive interpretation for the siloxane blend follows consequently this fine by arguing
that spatial restrictions in the order of trie coil d1nlensious are responsible for the exper1nlental observations. Since it is known front viscosity measurenlents that in the siloxane mixture the respective nlononler friction coefficients (A,B at the tenlperature of nleasuremeut are within a factor of 2.6 [8], we can exclude that trie reported results [6] are due to large diifereuces in trie nlicroscopic Rouse relaxation rates which enters trie Ronca nlodel. Thus the interpretation of spatial restrictions tentatively given in (6) seems to be feasible. In order to more quautitatively verify trie above speculations we present in this conlmunication a theoretical model based ou the variational treatment to account for the observed exper1nlentalfindiug. The latter is used to show that in conlpatible binary blends this kind of localization is important though it bas so for not been recognized nlainly due to the lack of expenmental data m the relevant t1nle t and
wave vector ranges. From our analysis we predict an anomalous static scattering behaviour
of conlpatible binary polymer blends as a function of the magnitude of the Flory-Huggins pa-
ran1eter xF conlpared to Ko " 2IN, trie value at trie spinodal. Trie implications of our results with respect to trie raudonl phase approx1nlatiou [9] will be discussed.
Results and discussion.
For trie physical nlodel we consider a single chain in trie synlnletric binary A/B polynler blend.
It has been shown [1] that a tagged chain (of the A type, say) in the blend can be described
by the following Harniltouian in dimensionless uuits
i NA ôJ~~ ~ NA NA
H =
/
~ ds +
/
ds/
ds'U(Ri (s) Ri (s')) (1)2 0 3 0 0
where U(r) is trie effective monomer-monomer potential. Trie Fourier transform of U(r) to
k-space is given by [1]
~~~~
l + VAAS(~(k) ÎÎ~ÎÎÎÎÎÎ(ÎÎS[~ÎÎVAAVAB Vi~) ~~~
The first part of equatiou (1) defines trie chain entropy [10], Ri (s) trie chain variable and s
the contour variable, which measures the distance along the chain. The effective monomer-
n1onomer potential U(k) is entirely defiued by trie bare structure factors of trie differeut species
S°(k), a, T
= A,B and the Flory xF-Parameter [9]. The forai of the interaction potential equation (2) is too conlplicated for analytical treatnlent. For the purpose in this paper it is sufficient to study a symnletric blend, 1-e- Si~(k)
= S[~(k) and VAA # VBB " V, VAB "
V+xF S(~(k) is the unperturbed (bare) structure factor which is given by the Debye function,
but well approximated by Si~(k)
= ~
The potential from equation (2) con therefore
1+ jk N
be rearranged to beconle in the dimensional uuits chosen above
v j(1/S°(k)) 2XF)
j3a)
Ul~) "
i + S°ik)v ((2/S°(k)) 2xF)
After algebraic rearrangenlents this is rewritten as:
~ ~~~ v(12 (2xF)~
~
" "
(k~ + 1/fi) (k~ + 1/f~) ~~
where v is the overall (bare) excluded volunle interaction of order 1 in the units chosen. Here (E is the Edwards screening length, (j~
r~ vc, c is the overall nlonomer concentration, aud (
is the correlation length of the critical fluctuation, 1-e- (~~
r~ 2(xo xF). xo
" 2/N is the
value of the interaction pararneter at the (nlean field) spinodal and ( the meau field correlation
length (~~
= 2(xo -xF). Trie approximation, equation (3) turns equation (2) into a long range
interaction potential near trie critical point Ko
" xF.
To investigate a theoretical backgrouud of the exper1nlents [6] a variational calculatiou of
an effective localization of the polynler in a favorable environnlent is perfornled. To see this
physically, 1nlagine fluctuations of a range over the correlation length ( = (Ko xF)~l/~ in
a binary bleud. The value of ( determines the spatial extension of already phase separated regions. When ( is very small, 1-e- of the order of the size of a monomer the blend is in the one
phase region, and the chains are roughly behaving as in a one component melt. In this case the chains are practically not localized as the volume occupied by A Inonomers is of the order of (~, where d is trie dimension of space (Fig. l). At small ( the melt eifect, 1-e- the first two
terrils of equation (3b) dominate and nothing peculiar happens.
~~ ~
( small, no localization ( langer, monomers staff tu Orly monomers of the saine
d~~i~ ~~ ~~~1~~ ~f j type are likely to be found
at scales (
Fig. l. Trie droplet picture of polymer mixtures. When trie correlation length of trie mixtures is of trie order of the chain length trie chains can become trapped inside regions of size x. This is trie
assumption of trie variational principle put forward in ibis paper.
As ( becomes langer by decreasing the temperature the polymer blend "phase separates" at
larger length scales (. Eventually ( becomes coInparable to the chain size Ni/~ l itself. At this temperature n r~
N~/~~l chains
occupy the correlation volume (~
r~
N~/~l~ at density equal
to one. For tllermodynanlic considerations, these cllains do not escape from this volunle easily,
1-e- tlley con be considered to be localized. Further decrease of tlle teInperature yields larger
and forger ( aud the chains move again freely in the voluIne (~
r~ O(Q), where Q is tlle volunle of tlle entire system.
It is therefore reasonable to assume localization of the chains within tlle correlation volunle
(~. TO defiue this process nlore precisely we perforIn a variational treatnlent for tlle tagged
chain Hamiltonian equation il). In the spirit of reference [3, 11] a harmonic triai potent1al is added and subtracted s1nlultaneously to equation (1)
V
= q(
~
R~(s)ds (4)
2
988 JOURNAL DE PHYSIQUE I N°7
The variational partition function is thus given by
Zvar = DR(s) xp1-Ho([R(s)])
+ ~q( dsR~(s) ds ds'U(R(s) R(s'))
Î
2 ~ ~ ~ ~Î(5)
where
~ ~ ~
~° ~~
(Î~
~
Î
~~~~~~~ ~~~
and )o "
) Il
)e~~°. Trie vanational paraIneter is qo, 1-e- the strengtll of the harmonico
potential. The inverse qp~ deternlines the range of localization of the chains. qo is determined
finally by minimizing Fvar
= logZvar with respect to qo.
As the original forIn of the effective potential between the monomers is long ranged (see Eq.
(2)) reasonable values for qo can be expected. For the analytical computations the form (3) for the potent1alis used. Trie diiferent averages in equation (5) are calculated straightforwardly
and after minimization of trie variationalfree energy the dominant contribution to qo is
~o ~3
v~ +lx~~~ 1+
Il
(7)Equation (7) is in particular interesting as it contains the pure melt eifect, i e. tlle first term in the bracket and tlle blend eifect expressed in the 1/(-terIn.
Tlle variational approach works better for forger (, since trie potent1al becomes Inore and
nlore long ranged. Trie "melt" effect N/(E is a fixed value as trie overall concentration does
not change by variation of trie tenlperature, 1-e- it is an irrelevant constant for our purpose.
Tllerefore we rewrite equation (7) as
~° = ~?
+al
18)wllere o is a constant. The localization eflect due to critical droplets [12] is given by qi m a
)
18a)It is interesting to note that individual polynler chains are localized in critical droplets and the localization vanishes at the cntical point. We therefore conclude that trie chains are localized already above tlle phase separation. The eflect of localization bas its nlaxinlunlwllen the size of the chains are of the order of the critical droplet, 1-e- R
r~ ( or (
r~
/Ù. The corresponding
value of 2xF is for this situation
~~~ N ~~~
which is precisely the xF value where the effective potential due to the blend eifect changes sign (see Eq. (3a)). At the critical point, Ko " xF,( diverges, and the localization vanishes.
This is physically clear, as the fluctuation are very large and the systeIn phase separates.
Using these results and the theory for Rouse dynanlics of spatially constraint systenls (pre-
sented in [6]) the nleasurenlents of reference [1] can be interpreted. The effective chain Hamil- tonian is given by
Hetr - £~d3
Ill
~+ l~i~£~R~(3) (1°)
The dynanlics and the structure factors of such chain has been worked out in detail in reference I?i and the coherent dynanlic structure factor shows a finite value for long t1nles t ~ cc, which
has been observed exper1nlentally Ill.
These results have consequences in the static scattering, also and an alternative nleasurenlent of the localization eifect induced by critical fluctuations is suggested. The collective structure
factor of a binary blend is usually analyzed in terrils of de Gennes (Gaussian) randonl phase approximation (RPR) which is for entirely symmetric blends given by
ù
= 2
sol~~
xfl(11)
where S°(k) is the structure factor of the ideal (unperturbed) chain. When the non perturba-
tive eifects of the localization are taken into account S°(k) has to be replaced by this one that is calculated by the effective Hamiltonian H~~, 1-e-
Si(k, q()
=
/~
ds/~
ds' e~~(~(~~~~~~'~~)(12)
o o
where the average is taken over the probability
exp (-H~~r)
Z
A reasonable approximation is given for the structure factor on a system with equation (10) together with trie use of (11) yields tue Kratky-plot of S(k) given in figure 2. It shows a peak at large localization and tue peak disappears near trie cntical point when trie value of
q( is zero. Therefore two independent possibilities of an experimental proof of trie fluctuation
induced localisation can be put forward. Trie first one is indicated by trie scattenng reported
in reference [6] and trie second one is an anomalous static scattering as indicated in figure 2.
The localisation eifects are temperature dependent and vanish at trie critical point. Finally
we would like to mention that such a non perturbative locahzation eifect is only observable in
polymer blends and not in Iow molecular weight systems.
k~S(k)
qo~ > °
qj =o
k
Fig. 2. Static scattenng from slightly localized chains. The hump m the Kratky plot is an effect of the localization.
On trie other hand a large amount of very reliable simulation data of Binder and cc-workers exist [14-16]. In our earlier study (Ref. [l]) we had predicted trie eifect of chain shrinking before the macrophase separation of the polymer blend using trie effective (meanfield) potential from
990 JOURNAL DE PHYSIQUE I N°7
equations (1-3). Indeed such eifects are beyond trie classical RPR assumption and have been confirmed by Sariban and Binder [14]. The chain shrinking, which we assigned in reference [1]
to fluctuation eifects, can now be understood in another physical way. When the A chains are localized in droplets of size (, they are compressed by the presence of trie B chains outside the
droplet. The picture of chain shrinking has been supported aise in a diiferent approach Il?]
and moreover by a recent renormalization study [18]. The other eifects that are predicted in
the present study bave not been investigated by the Binder group explicitly. A precise study of the collective structure factor seem to be very computing time expensive [14-16], moreover
the simulation of trie dynamic scattering function S(k,t) has not been carried out yet. These data would be very useful for computational support for our present predictions in polymer blends.
As a final remark we would like to mention that the localization effect predicted in this paper con possible shed light on another result by Kutner et al. [19]. These authors discuss the diffusion in concentrated Iattice gases of particles with attractive nearest neighbour inter- actions. They find a vanishing part in tue diffusion constant near the critical point. Such
a behaviour can be viewed together with the localisation picture. It is indeed interesting to
compute the diffusion constant for the polymer mixture near the critical point and investigate the connection with Kutner's result.
Acknowledgements.
T.A.V. acknowledges very useful conversations with Kurt Binder on his work and the present
state of the art of the simulation of polymer blends and for bringing reference [19] to his
attention.
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